1276 

 From this we find 



hx(f{x) = / 1 K{x,y) L,,<f{y)dy - 1 



a 



Substituting in this equation 



b 



,f\t]) = X\ K{n,y)(i{y)dy, 



pin) \K{x,ii)(p'{i]) — r/(i;) 



b 



. rdK(ti,y) 



(t 



we get (13). 



Theorem LI. The necessary and sufjiclent condition that a 

 complete system of orthogonal characteristic functions of K{a',y) be 

 solutions of (S), is that K{x,y) satisfies identically in x and y the 



equation 



A(.t-,v) = . (14) 



l^roof. First we assume A(,i\7) = 0. If theiM/(:}') be a characteristic 



luiiction belonging to tlie chaiacteiislic number /, it follows from 



(13) that 



Lr'i{.v) = KiK{x,y)lly(fiy)dy. 



Consequently A.^ r/^^.r) is a characteristic function for the nianber ^.. 

 Now if 'fi(.r)> •• • •> V"('^') ^^ ^ complete orthogonal ami noi-malized 

 system of characteristic functions all belonging to /, the functions 



£^_^i^ (.,.)^ L,ifn{x) also will be characteristic functions for the 



value P.; consequently they are expressible in (i\{.r), . . . ., ipn{x) by 

 formulae of the form 



L,ifi{x)z= ^ ci,ipj{x) . . . . . . (15) 



From this 



5y = I y.;(^")A. 



qi (x)da:. 



The formula of Green now gives 



b 



^ij <^ii — 



,■ = I {(pj{a;)A,(f{ (.r)— (fi {x)L:,q j{x)\da: 

 a 



\p{ri)\<fAh)'/'iiri) - ^r'inh'AnYi]^ 



